Mechanical work
From Wikipedia, the free encyclopedia.
Work (abbreviated W) is the energy transferred by a force to a moving object. Work is a scalar quantity, but it can be positive or negative. It is associated with a change in energy, but not all changes in energy can be readily analysed in terms of work.
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Definition
Note: Readers not familiar with multivariate calculus or vectors, please see "Simpler formulae" below
Work is defined as the following line integral
where
- C is the path or curve traversed by the object;
is the force vector;
is the position vector.
Units
The SI derived unit of work is the joule (J), which is defined as the work done by a force of one newton acting over a distance of one metre. The dimensionally equivalent newton-meter (N·m) is sometimes used instead; however, it is also sometimes reserved for torque to distinguish its units from work or energy.
Non-SI units of work include the erg, the foot-pound, and the foot-poundal.
Simpler formulae
In the simplest case, that of a body moving in a steady direction, and acted on by a force parallel to that direction, the work is given by the formula
where
- F is the force and
- s is the distance traveled by the object.
The work is taken to be negative when the force opposes the motion. More generally, the force and distance are taken to be vector quantities, and combined using the dot product:
where φ is the angle between the force and the displacement vector.
This formula holds true even when the force acts at an angle to the direction of travel. To further generalize the formula to situations in which the force and the object's direction of motion changes over time, it is necessary to use differentials, d, to express the infinitesimal work done by the force over an infinitesimal displacement, thus:
The integration of both sides of this equation yields the most general formula, as given above.
Types of work
Forms of work that are not evidently mechanical, such as electrical work, can be considered as special cases of this principle; for instance, in the case of electricity, work is done on charged particles moving through a medium.
Heat conduction from a warmer body to a colder one is not normally considered to be a form of mechanical work, because at the macroscopic level, there is no measurable force. At the atomic level, there are forces as the atoms collide, but they average to nearly zero in bulk.
Not all forces do work. For instance, a centripetal force in uniform circular motion does not transfer energy; the kinetic energy of the object undergoing the motion remains constant. This fact is confirmed by the formula: if the vectors of force and displacement are perpendicular, their dot product is zero. This does not mean the centripetal force is doing nothing; the velocity, and hence the momentum of the object undergoing circular motion, is changing continuously. Momentum is a vector quantity, change of direction is a change of momentum.
Mechanical energy
The mechanical energy of a body is that part of its total energy which is subject to change by mechanical work. It includes kinetic energy and potential energy. Some notable forms of energy that it does not include are thermal energy (which can be increased by frictional work, but not easily decreased) and rest energy (which is constant so long as the rest mass remains the same).
Conservation of mechanical energy
The principle of conservation of mechanical energy states that, if a system is subject only to conservative forces (e.g. only to gravitational force), its mechanical energy remains constant.
For instance, if an object with constant mass is in free fall, the total energy of position 1 will be equal position 2.
where
- Ek is the kinetic energy, and
- Ep is the potential energy.
References
- Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.), Brooks/Cole. ISBN 0534408427
- Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.), W. H. Freeman. ISBN 0716708094
